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Full text of "Men Of Mathematics"

ANIMA CANDIDA

on which the multiform function is uniform, that is, on which
to each 'place* on the surface corresponds one, and only one,
value of the function represented.

Riemann united, as it were, ail the n planes into a single plane,
and he did this by what may at first look like an inversion of the
representation of the n branches of the n-valued function on n
distinct planes; but a moment's consideration will show that,
in effect, he restored uniformity. For he superimposed n s-planes
on one another; each of these planes, or sheets, is associated
with a particular branch of the function so that, as long as z
moves in a particular sheet, the corresponding branch of the
function is traversed by re (the n-valued function of z under
discussion), and as z passes from one sheet to another, the
branches are changed, one into another, until, on the variable z
having traversed all the sheets and having returned to its
initial position, the original branch is restored. The passage of
the variable z from one sheet to another is effected by means of
cuts (which may be thought of as straight-line bridges) joining
branch points; along a given cut providing passage from one
sheet to another, one 'lip' of the upper sheet is imagined as
pasted or joined to the opposite lip of the under sheet, and
similarly for the other lip of the upper sheet. Diagrammatically,
in cross-section,

Upper   $k**+

Lover &ģet                                LQHCT
The sheets are not joined along cuts (which may be drawn IB
many ways for given branch points) at random, but are so
joined that, as -z traverses its n-sheeted surface, passing from
one sheet to another as a bridge or cut is reached, the analytical
behaviour of the function of z is pictured consistently, parti*
cularly as concerns the interchange of branches consequent on
the variable z* if represented on a plane, having gone completely
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